High-throughput calculation screening for new silicon allotropes with monoclinic symmetry

High-throughput calculation screening is presented for 87 new low-energy silicon allotropes with monoclinic symmetry.


Introduction
Monocrystalline silicon is still the cornerstone of the modern semiconductor and microelectronics industry, but some physical properties of diamond Si limit its application, such as its indirect band gap (Kim et al., 2015). This limitation renders silicon undesirable for the next generation of efficient platforms for applications such as high-performance transistors (Theis & Solomon, 2010), light-emitting diodes (Ng et al., 2001;Fujita, 2013) and thin-film photovoltaic devices (Botti et al., 2012). Therefore, searching for silicon allotrope structures with a direct or quasi-direct band gap remains a focus for researchers.
Computer-aided materials science has been widely used in the field of electronic materials simulation (Wang et al., 2021;Wei et al., 2023;Zhang et al., 2022) and there have been many theoretical reports on the design of new silicon materials (Lee et al., 2014;Su et al., 2022;Song et al., 2022;He et al., 2018;Fan et al., 2023;Cui et al., 2023;Wei et al., 2022;Cheng et al., 2018) based on density functional theory (Hohenberg & Kohn, 1964;Kohn & Sham, 1965). However, most of the new silicon structures still belong to the class of indirect band gap semiconductor materials (Cui et al., 2023;Wei et al., 2022;Cheng et al., 2018). Kim et al. (2015) reported the discovery of a new orthorhombic silicon allotrope, Si 24 , formed through a novel Published under a CC BY 4.0 licence high-pressure precursor process, and found that it possesses a quasi-direct band gap near 1.3 eV. Subsequently, Linghu et al. (2017) attempted to modulate Si 24 into a direct band gap semiconductor by doping with B, Al, Ga, P and As atoms. However, after doping, the results were not satisfactory due to the quasi-direct band gap semiconductor material produced. In other words, doping with Group III or V elements did not lead to successful modulation of Si 24 into a direct band gap material, but the band gap range of doped Si 24 remains the best for making solar cells. Similar electronic band gap modulation can also be achieved by doping with homologous elements Fan et al., 2019;Song et al., 2019;Fan et al., 2020;Wang et al., 2017;Fan et al., 2022) but there have been few successful cases (Wang et al., 2017;Fan et al., 2022).
The traditional methods of discovering new materials represented by the empirical trial-and-error method and density functional theory (DFT) (Hohenberg & Kohn, 1964;Kohn & Sham, 1965) cannot keep up with the development of today's materials science because of their long development cycle, low efficiency and high cost (Curtarolo et al., 2013). High-throughput computational methods are widely used in materials design, testing and analysis, and in other fields, because of their low computing cost, short development cycle, strong data processing ability and high prediction performance. High-throughput computation can greatly reduce the computing cost and shorten the development cycle. At present, a large number of materials design reports (Olsen et al., 2019;Zhao et al., 2023;Shi et al., 2021;Sun & Schwingenschlö gl, 2021;Wei et al., 2020;Al-Fahdi, Rodriguez et al., 2021;Zhang et al., 2018;Blatov et al., 2021;Takagi & Maeda, 2020) utilize highthroughput computations.
Recently, the configuration space of two-dimensional planar sp 2 carbon was studied using high-throughput calculations and the RG 2 method (a random strategy based on a combination of group and graph theory) . To be specific, 1114 new carbon allotropes were identified , including 241 semiconductors, 683 normal metals and 190 Dirac semimetals, and a set of crystal structure search code using RG 2 methods was proposed by Shi et al. (2018), which can be used for research into crystal structure solution, crystal structure prediction and design of functional materials of interest in condensed matter physics and computational materials science. Other structure prediction methods or programs such as basin hopping (Wales & Doye, 1997;Iwamatsu & Okabe, 2004;White & Mayne, 1998), USPEX (Oganov & Glass, 2006;Glass et al., 2006;Oganov et al., 2010;Lyakhov et al., 2010;Oganov et al., 2011;Lyakhov et al., 2013;Niu et al., 2019) and CALYPSO (Wang et al., 2010;Wang et al., 2012) also play a significant role in ab initio structure prediction.
In this work, 87 monoclinic silicon allotropes are identified using high-throughput calculations and RG 2 . All 87 of these novel allotropes have lower relative enthalpy values (below 0.300 eV per atom) than diamond Si. According to their electronic band structures, 13 of these new silicon allotropes have a direct or quasi-direct band gap and 12 of them have metallic characteristics.

Computational methods
The space group range we set in RG 2 was 3-15 for monoclinic symmetry; RG 2 randomly generated 15 669 initial structures with lattice parameters of 3 Å a 20 Å , 3 Å b 30 Å , 3 Å c 50 Å , 30 150 , 30 150 and 30 150 . After duplicate structures had been filtered out by RG 2 , a total of 1015 structures of silicon allotropes with monoclinic symmetry remained. Structures that were obviously twodimensional were then removed, leaving only $4.53% (710) of the original 15 699 that can be built into 4-coordinated networks (4-connected quotient graph), which is slightly fewer than obtained by Shi et al. (2021) ($5%). We only ran the process of predicting the structures for 60 min and the speed of generating new structures is roughly quadratically related to the number of structures. As shown in Fig. 1(a), the generation rate decreases gradually with time. Approximately 710 new three-dimensional silicon structures are in the range of space groups 3-15; that is, approximately 69.95% (of the 1015 monoclinic structures) meet the requirements. Among all the structures predicted by RG 2 , the space group C2/m (No. 12) is the most common. All the space groups generated by RG 2 and the number of structures belonging to each space group are shown in Figs. 1(b)-1(d). Then, removing duplicate structures from the set of proposed structures leaves 389 different 3D sp 3 monoclinic silicon allotropes.
Density functional theory (DFT) (Hohenberg & Kohn, 1964;Kohn & Sham, 1965) based on the Cambridge series total energy package (CASTEP; Clark et al., 2005) was used to implement this work. RG 2 generated thousands of geometrically acceptable structures and these were then optimized using CASTEP. The Perdew-Burke-Ernzerhof (Perdew et al., 1996) functional of the generalized gradient approximation (GGA) exchange and correlation functional were used for geometry optimization and property prediction. To describe the interaction between electrons and ions, the ultrasoft quasipotential (Vanderbilt, 1990) was used in the calculations. The Broyden-Fletcher-Goldfarb-Shanno (Pfrommer et al., 1997) minimization was used to optimize the geometry. For all monoclinic Si allotropes, the plane wave energy cut-off value was set to 340 eV. The Brillouin zone was sampled with Monkhorst-Pack meshes (Monkhorst & Pack, 1976) with special k-point grids ($2 Â 0.025 Å ) for all monoclinic Si allotropes.
The naming rules for the new structures, taking 8-6-24-232931 as an example, are as follows: the first component 8 of the crystal structure name is the space group number, the second component 6 is the number of non-equivalent positions of silicon atoms in the conventional cell, the third component 24 is the number of atoms in the conventional cell and the fourth component 232931 is an anti-duplication number. Here, a random number representation is used to generate the corresponding initial structure, which ensures that different structures will have different numbers under the same input parameters, and also ensures that the probability of repeating a number for different structures under different input parameters is very low.

Structural and topology properties
Because of the variety and importance of 3D silicon allotropes, we have revisited the structural configuration of 3D sp 3 silicon allotropes in a more systematic way using a highthroughput methodology associated with RG 2 . As shown in Fig. 2, we first collected the previously reported silicon allotropes and then learnt their fundamental structural features using RG 2 to build the initial database.   Finally, appropriate structures were generated by limiting the coordination numbers of the atoms, the bond length, the bond angle, the space group and other structural information.
First, as shown in Fig. 2, we used CASTEP to optimize the 389 different 3D monoclinic sp 3 silicon allotrope structures. Through structural optimization by CASTEP, some structures become other phases. For instance, some structures are transformed into a variety of hexagonal phases, as proposed recently by Wei et al. (2022), and diamond polytypes, such as 15-2-16-231721, 12-4-16-030405 and 15-2-16-001019, are transformed into 4H or 2H Si structures, 5-6-24-163555 is transformed into a 6H structure, 12-7-28-083128 is transformed into a 7H structure, 15-3-24-070924 is transformed into a 9R structure and 3-4-32-31817 is transformed into a diamond polytype. As a result, 124 structures were able to maintain a monoclinic phase. We then used CASTEP to verify the elastic constants. It was found that there are many silicon allotropes which fail to meet the condition of mechanical stability. The final destination of these 389 3D monoclinic sp 3 silicon allotropes after CASTEP optimization is explained in the flow chart in Fig. 2, and all the results are shown in Fig. S1 in the supporting information. Since there are many structures in space groups 12-15, the screening process is complex, and the specific conversion process is shown in Figs. S1(j), S1(k) and S1(l). Taking space group C2/c (No. 15) as an example, there are 126 structures in space group C2/c in total. After structural optimization using CASTEP based on DFT, 12 structures are repeated, nine structures are scattered (unstable) and three are transferred to space group No. 193. A detailed structural transformation is shown in Fig. S1(l). Finally, 20 structures in space group C2/c were obtained.
After the above geometric structure optimization, elastic constant and phonon spectra calculations were done, also using CASTEP. There are 87 different new 3D silicon allotrope structures with monoclinic symmetry (excluding those with structural deformation or space group transformation) whose structures are different from those of monoclinic silicon allotrope structures predicted previously (Wang et al., 2014;Fan et al., 2016;Wei et al., 2019;Lee et al., 2016;Amsler et al., 2015). We further studied their structures and physical properties using first-principles calculations. Among them, in terms of electronic band structures, all of the space groups P2 1 , C2, Cm, P2/m and P2 1 /m are semiconductor structures, while the space groups C2/m, P2/c, P2 1 /c and C2/c have both semiconductor and metal structures. The silicon allotropes in P2/c show the most metallic structures, up to six, while space group C2/m shows the fewest with metallic properties, having only one structure. Through electronic band structure calculations, we found that among the 87 different new 3D monoclinic sp 3 silicon allotrope structures, there are 12 with metal properties and 75 with semiconductor properties.  Table 1 The crystal lattice parameters (a, b, c in Å ; in ) and formation energies (eV atom À1 ) of monoclinic silicon allotropes with direct and quasi-direct band gaps and diamond Si.  (1994), experimental data.

Figure 2
The work flow for high-throughput screening of the crystalline structures and electronic properties of Si allotropes in 3-15 associated with RG 2 .
The topological characteristics of all the monoclinic silicon allotrope structures were calculated using ToposPro (Blatov et al., 2014) on the Topcryst website (https://double.topcryst. com/index.php) (Shevchenko et al., 2022). Among the 87 structures predicted in this work, the topological types of 13 of them are previously unknown. In addition, only a few topological types of silicon structures are similar to those recorded in SACADA (Samara Carbon Allotrope Database; Hoffmann et al., 2016), such as 4 2 T112 which is the same as the 147th in SACADA, and 4 2 T265 which is the same as the 442nd in SACADA. Although these same topology types are recorded in SACADA, there are still more than 50 silicon allotropes not recorded in SACADA, with orders 22-74 as shown in Table S1.

Stability
It is important to analyse the stability of new materials to understand them better. Therefore, the elastic constants of 13 new silicon allotropes with direct or quasi-direct band gap are listed in Table 2, and the elastic constants of the other silicon allotropes are shown in Table S2. For stable monoclinic phases, their independent elastic constants C 11 , C 22 , C 33 , C 44 , C 55 , C 66 , C 12 , C 13 , C 15 , C 23 , C 25 , C 35 and C 46 should obey the following generalized Born mechanical stability criteria (Wu et al., 2007):  Table 2 Elastic constants (GPa) and elastic moduli (GPa) of monoclinic silicon allotropes with direct and quasi-direct band gaps, and diamond Si.

Figure 3
The crystal structures of the five novel monoclinic silicon allotropes with direct band gaps.
g ¼ C 11 C 22 C 33 À C 11 C 2 23 À C 11 C 2 23 À C 11 C 2 23 þ 2C 12 C 13 C 23 : Table 2 and Table S2 show that all the calculated elastic constants meet the mechanical stability criteria, and thus all the silicon allotropes mentioned in this work are mechanically stable. The crystal densities and formation energies for all 87 new 3D sp 3 silicon allotrope structures are shown in Fig. 4(a). The formation energies of all allotropes were calculated by setting the energy of diamond Si to 0. The detailed crystal densities and formation energies for the 13 new silicon allotropes with direct or quasi-direct band gaps are shown in Fig. 4(b). These monoclinic silicon structures with direct or quasi-direct band gaps are mainly clustered in four space groups, P2/m, P2 1 /m, C2/m and C2/c.
The relative energies of these novel monoclinic silicon structures with direct or quasi-direct band gaps are in the range 0.06-0.18 eV per atom; the relative energies of most of the structures are less than 0.30 eV per atom. Fig. 4(a) shows the crystal densities of all 87 new 3D sp 3 silicon allotropes, which range from 1.50 to 2.50 g cm À3 . In addition, from Figs. 4(a) and 4(b), the formation energies of these silicon structures are 85-249 meV per atom, higher than that of diamond Si.
The dynamic stability of these novel monoclinic silicon allotropes was also investigated. The phonon spectra are shown in Fig. S3. No negative frequencies are found throughout the first Brillouin region, indicating that these new monoclinic silicon allotropes are dynamically stable.

Mechanical properties
Table 2 also shows the bulk modulus, shear modulus and Young's modulus of 13 new silicon allotropes with direct or quasi-direct band gaps and of diamond Si. The bulk moduli for 13 new silicon allotropes with direct or quasi-direct band gaps are in the range 63-80 GPa, and the shear moduli are in the range 27-55 GPa. The other elastic constants and the bulk modulus, shear modulus and Young's modulus are shown in Table S2. From Table 2 and Table S2, the bulk moduli of these monoclinic silicon allotropes range from 42 to 99 GPa, and the shear moduli range from 27 to 66 GPa. The mechanical properties of some of these new structures are better than those of diamond silicon. The silicon allotrope 13-3-12-232508 has the greatest bulk modulus and 15-3-24-232448 has the greatest shear modulus. The bulk moduli of three new structures (13-3-12-232508, 14-5-20-232541 and 15-3-24-232448) are greater than that of diamond Si. The shear moduli of two new structures (14-4-16-232446 and 15-3-24-232448) are greater than or equal to 60 GPa, and thus very close to the value for diamond Si.
The 3D directional dependence of the Young's modulus for an isotropic structure can be illustrated as a sphere, while deviation from a sphere represents anisotropy (Xing & Li, 2023a,b;Yu et al., 2022;Liu et al., 2023). Therefore, to observe the anisotropy of these novel monoclinic silicon allotropes with direct or quasi-direct band gaps more directly, the 3D directional dependence of the Young's moduli for the 13 silicon allotropes with direct and quasi-direct band gaps were studied, and the results are shown in Fig. 5 and Fig. S4. All the 3D directional dependences of the Young's moduli deviate from spheres, so they all show different degrees of anisotropy.
The effective mass of the carrier has a great influence on the transport performance of a semiconductor material. To understand and study the transport properties of these silicon allotropes with direct band gap more directly and effectively, their effective masses were studied. The effective mass (m) can be calculated as follows (Bardeen & Shockley, 1950): The 3D directional dependence of the electron and hole effective masses of 11-4-16-232321 and 11-5-20-234816 are shown in Fig. 7. Table 3 lists the hole effective masses of 11-4-16-232321, 11-5-20-234816 and diamond Si. As seen in Fig. 7, all the 3D shapes of the hole effective masses of 11-4-16-232321 and 11-5-20-234816 exhibit anisotropy. As given in Table 3, the theoretical values for the hole effective masses for diamond Si are in excellent agreement with the experimental results for heavy holes. The hole effective masses of 11-4-16-232321 and 11-5-20-234816 are slightly larger than that of diamond Si. The electron effective masses of 11-4-16-232321, 11-5-20-234816, 12-2-16-231721, 12-3-12-232409 (mC12-Si) and diamond Si are listed in  Three-dimensional plots of the Young's moduli for the four new monoclinic silicon allotropes with direct band gaps.   -4-16-232321, 11-5-20-234816, 12-2-16-231721 and 12-3-12-232409 (mC12-Si) are smaller than that of diamond Si, with the value for 11-5-20-234816 being approximately twothirds that of diamond Si, the values for 11-4-16-232321 and 12-2-16-231721 being less than one-third of that of diamond Si and the value for 12-3-12-232409 (mC12-Si) being approximately one-third that of diamond Si. In addition, the electron effective masses m t2 of 11-4-16-232321 and 11-5-20-234816 are slightly smaller than that of diamond Si. The absorption spectra for the new monoclinic silicon allotropes with direct and quasi-direct band gaps are shown in Fig. 8, and the absorption spectra for the new monoclinic The electron band structures of the 13 novel monoclinic silicon allotropes with direct or quasi-direct band gaps. silicon allotropes with indirect band gaps are shown in Fig. S7. As can be seen in Fig. 8 and Fig. S7, the absorption spectra for these monoclinic silicon allotropes with direct and quasi-direct band gaps are stronger than that of diamond Si in the visible region. From Fig. 8(b), it can be observed that the monoclinic silicon allotropes with quasi-direct band gaps show better photon absorption performance than diamond Si, which is due to the fact that all their absorption spectra in the visible range have a higher magnitude than that of diamond Si. These results also suggest that our proposed silicon allotropes with quasi-direct band gaps start to absorb sunlight at lower energies than diamond Si, as do those with direct band gaps proposed in this work. The novel silicon allotropes with indirect band gaps also have better light absorption capacity than diamond Si. From what has been discussed above for Fig. 8 and Fig. S7, it is concluded that these monoclinic silicon allotropes show promising prospects for photovoltaic applications.

Conclusions
In total, 87 new monoclinic silicon allotropes have been obtained and confirmed by RG 2 and high-throughput calculations. The vast majority (more than 86%) of these allotropes are in space groups C2/m, P2/c, P2 1 /c and C2/c. Crystal structures in space group P2/c are the greatest in number, while crystal structures in space groups P2, Pm and Cc were not found in this work. This is probably due to the fact that we only used RG 2 for 60 min, which is not enough time to generate more structures in space groups P2, Pm, Pc and Cc.
All the elastic constants for the new monoclinic silicon allotropes proposed in this work meet the Born mechanical stability criteria. Among the 87 allotropes presented here, 13-3-12-232508 silicon shows the greatest bulk modulus (99 GPa) and 15-3-24-232448 silicon shows the greatest shear modulus (66 GPa).
All 74 novel monoclinic silicon allotropes with semiconductor properties have a strong absorption capacity in the visible region, which shows great potential for application in optoelectronic devices.

Figure 8
The absorption spectra for monoclinic silicon allotropes with (a) direct and (b) quasi-direct band gaps. The absorption spectra were calculated using the HSE06 hybrid functional.